Character table for point group D8d

D8d E 2S16 2C8 2(S16)3 2C4 2(S16)5 2(C8)3 2(S16)7 C2 8C'2 8d
linear functions,
rotations
quadratic
functions
cubic
functions
A1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 - x2+y2, z2 -
A2 +1 +1 +1 +1 +1 +1 +1 +1 +1 -1 -1 Rz - -
B1 +1 -1 +1 -1 +1 -1 +1 -1 +1 +1 -1 - - -
B2 +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1 z - z3, z(x2+y2)
E1 +2 +2cos(1/8) +(2)½ +2cos(3/8) 0 -2cos(3/8) -(2)½ -2cos(1/8) -2 0 0 (x, y) - (xz2, yz2) [x(x2+y2), y(x2+y2)]
E2 +2 +(2)½ 0 -(2)½ -2 -(2)½ 0 +(2)½ +2 0 0 - (x2-y2, xy) -
E3 +2 +2cos(3/8) -(2)½ -2cos(1/8) 0 +2cos(1/8) +(2)½ -2cos(3/8) -2 0 0 - - [x(x2-3y2), y(3x2-y2)]
E4 +2 0 -2 0 +2 0 -2 0 +2 0 0 - - -
E5 +2 -2cos(3/8) -(2)½ +2cos(1/8) 0 -2cos(1/8) +(2)½ +2cos(3/8) -2 0 0 - - -
E6 +2 -(2)½ 0 +(2)½ -2 +(2)½ 0 -(2)½ +2 0 0 - - [xyz, z(x2-y2)]
E7 +2 -2cos(1/8) +(2)½ -2cos(3/8) 0 +2cos(3/8) -(2)½ +2cos(1/8) -2 0 0 (Rx, Ry) (xz, yz) -


Additional information

Number of symmetry elements h = 32
Number of irreducible representations n = 11
Abelian group no
Subgroups Cs , C2 , C4 , C8 , D2 , D4 , D8 , C2v , C4v , C8v , S16
Optical Isomerism (Chirality) no


Reduction formula for point group D8d

Type of representation

general 3N vib

E 2S16 2C8 2(S16)3 2C4 2(S16)5 2(C8)3 2(S16)7 C2 8C'2 8d




Multipoles

dipole (p) B2+E1
quadrupole (d) A1+E2+E7
octopole (f) B2+E1+E3+E6
hexadecapole (g) A1+E2+E4+E5+E7
32-pole (h) B2+E1+E3+E4+E5+E6
64-pole (i) A1+E2+E3+E4+E5+E6+E7
128-pole (j) B2+E1+E2+E3+E4+E5+E6+E7
256-pole(k) A1+B1+B2+E1+E2+E3+E4+E5+E6+E7
512-pole (l) A1+A2+B2+E1+E2+E3+E4+E5+E6+2E7

First nonvanishing multipole: quadrupole

Literature



Character tables for chemically important point groups Computational Laboratory for Analysis, Modeling and Visualization Jacobs University Bremen